MARK KAC SEMINAR

In this talk I will discuss some recent results for Last Passage Percolation with general weights. We will introduce Busemann functions, show that they correspond to the equilibrium measures of a related interacting fluid process, and prove that these equilibrium measures are attractors for quite general initial conditions. Finally, I hope to give you some intuition on how these results could lead to a proof of the conjectured cube-root behavior of these systems. The proof would rely on how well the equilibrium measures can be approximated by Brownian motion with drift. Unfortunately, we have not been able yet to control these measures in a sufficient way.

This is joined work with Leandro Pimentel (Federal University of Rio de Janeiro).

The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress for the 4-dimensional self-avoiding walk has been based on a renormalisation group analysis. This analysis takes as its starting point an exact representation of the self-avoiding walk problem as an equivalent problem for a perturbation of a Gaussian integral involving anti-commuting variables (fermions). This lecture will give a self-contained introduction to fermionic Gaussian integrals and will explain how they can be used to represent self-avoiding walks.

Reference: D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys 6, pp. 34-61 (2009).